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title: W05 NotesVideos, Organic Chemistry Tutor
The pressure in a liquid is a function of the depth alone. This is a fundamental fact about liquids.
The fluid pressure in a liquid is a function of depth:
In SI units:
The pressure of a fluid acts upon any surface in the fluid by exerting a force perpendicular to the surface. Force is pressure times area. If the pressure varies across the surface, the total force must be calculated using an integral to add up differing contributions of force on each portion of the surface.
Total fluid force on plate:
Use
Have
Find the total force on the submerged vertical plate with the following shape: Equilateral triangle, sides
Solution
Videos, Math Dr. Bob:
The moment of a region to an axis is the total (integral) of mass times distance to that axis:
“Moment to
“Moment to
These formulas are obtained by slicing the region into rectangular strips that are parallel to the axis in question.
The area per strip is then:
The idea of moment is related to:
The center of mass (CoM) of a solid body is a single point with two important properties:
Note:
Coordinates of the CoM:
Here
Notice how these formulas work. The total mass is always
In other words, through the formula
Find the CoM of the region depicted:
Solution
A downside of the technique above is that to find
When the region lies between
When the region lies between
The idea for these formulas is to treat each vertical strip as a point concentrated at the CoM of the vertical strip itself.
The height to this midpoint is
Find the CoM of the region:
Solution
Compute the CoM of the region below
Solution
Two useful techniques for calculating moments and (thereby) CoMs:
Additivity says that you can add moments of parts of a region to get the total moment of the region (to a given axis).
A symmetry principle is that if a region is mirror symmetric across some line, then the CoM must lie on that line.
Find the center of mass of the region below.
Solution
Find the center of mass of the region which combines a rectangle and triangle (as in the figure) by computing separate moments. What are those separate moments? Assume the mass density is
Solution